He was one of the first to study differential equations involving complex variables, and his work led to a profound connection with group theory.
He introduced new general methods in the study of partial differential equations and applied them to produce the first major study of shock waves..
Bernhard Riemann, (born September 17, 1826, Breselenz, Hanover [Germany]—died July 20, 1866, Selasca, Italy), German mathematician whose profound and novel approaches to the study of geometry laid the mathematical foundation for Albert Einstein's theory of relativity..
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point)..
Riemannian Geometers also study higher dimensional spaces.
The universe can be described as a three dimensional space.
Near the earth, the universe looks roughly like three dimensional Euclidean space.
However, near very heavy stars and black holes, the space is curved and bent..
Uses of the Riemann tensor in cosmology As was mentioned before, Riemann's revolutionary ideas about the concept of physical space where given a definite form by Albert Einstein when he formulated the Theory of General Relativity, with the help of his, more mathematically minded, classmate and friend Marcel Grossmann7.
The Riemann integral is indeed a physically fashionable object, in particular the incremental version before the limit is taken, before an “increment” is transformed into a “differential”, which is a far more elusive concept indeed! (again, I know this quite well from my own students).
This is the Riemann hypothesis which remains today one of the most important unsolved problems of mathematics.
Riemann died of tuberculosis in 1866, while trying to recover from it in Selasca (Italy), surrounded by his wife and 3-year-old daughter.
He was only 39.
From his work, it is very clear that Riemann was interested in physics.
They were initially proposed by the German mathematician Bernhard Rie- mann around the year 1856, who was student of Gauss at that time.
Riemann was the first one to work on the hypothesis that space does not need to obey Euclid’s axioms.
That space is known today as Riemannian space.
